The typical example is something like headcount targets that lead to employing or laying off some of the wrong people.
I think that profit is more of a force like gravity than a future target that distorts strategy.
But in the general case I mostly agree with you. Sometimes though, businesses reach a market saturation level where “making the line go up” every quarter is bad for the health of the business.
profit is usually not the misplaced metric, because profit is the goal.
But if you have another goal such as well-being of the people of your country then you might set up GDP measures and call it a success when everyone is miserable. (I know GDP does not directly measure profit, this is in the nature of a real world example how something usually not thought of as a Goodhart's law measure might actually work)
Every economic systems aims to benefits society in general, and capitalism claims that (to great extent) free market allocation of capital is the best way to do that, using profit (or net increase in capital) as the optimizing function.
Well, when that measure becomes the target, it can get hacked, where profit can be maximised without getting closer to the intended goal. So you can make a very profitable company that just makes society worse. And alot of times that happens when these companies see regulations designed to limit their damage to society as just restrictions on their profitability, rather than understanding that the real goal of their company is to make society better, and profit is just the carrot hanging in front of their individual hamster wheel.